Triangles $ABC$ and $DEF$ are graphed in the $xy$-plane. Triangle $ABC$ has vertices $A,B,$ and $C$ at $(1,2),(1,6),$ and $(5,2),$ respectively. Triangle $DEF$ has vertices $D,E,$ and $F$ at $(1,2),(1,2+4m),$ and $(1+4m,2),$ respectively, where $m$ is a positive constant. If the measure of $\angle B$ is $x^{\circ },$ what is the measure of $\angle E$?